Evaluate the integral without using tables.

$\int}_{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}$

BenoguigoliB
2021-11-08
Answered

Evaluate the integral without using tables.

$\int}_{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}$

You can still ask an expert for help

yagombyeR

Answered 2021-11-09
Author has **92** answers

Step 1

Given,

$I={\int}_{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}$

Step 2

Formula used:

$\int \frac{dx}{\sqrt{1-{x}^{2}}}={\mathrm{sin}}^{-1}\left(x\right)+C$

$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$

Step 3

Apply the above formula, we get

$I={\int}_{0}^{1}\frac{dx}{\sqrt{1-{x}^{2}}}$

$={\left[{\mathrm{sin}}^{-1}\left(x\right)\right]}_{0}^{1}$

$={\mathrm{sin}}^{-1}\left(1\right)-{\mathrm{sin}}^{-1}\left(0\right)$

$={\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(\frac{\pi}{2}\right)\right)-{\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(0\right)\right)$

$=\frac{\pi}{2}-0$

$=\frac{\pi}{2}$

Given,

Step 2

Formula used:

Step 3

Apply the above formula, we get

asked 2022-04-06

I want to compute

${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}\frac{1}{\sqrt{x+yi+2}}dy$

where i is the imaginary number. How to compute this integral?

where i is the imaginary number. How to compute this integral?

asked 2021-06-12

Explain why each of the following integrals is improper.

(a)${\int}_{6}^{7}\frac{x}{x-6}dx$

-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.

-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.

-The integral is a proper integral.

(b)${\int}_{0}^{\mathrm{\infty}}\frac{1}{1+{x}^{3}}dx$

Since the integral has an infinite interval of integration, it is a Type 1 improper integral.

Since the integral has an infinite discontinuity, it is a Type 2 improper integral.

The integral is a proper integral.

(c)${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{x}^{2}{e}^{-{x}^{2}}dx$

-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.

-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.

-The integral is a proper integral.

d)${\int}_{0}^{\frac{\pi}{4}}\mathrm{cot}xdx$

-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.

-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.

-The integral is a proper integral.

(a)

-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.

-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.

-The integral is a proper integral.

(b)

Since the integral has an infinite interval of integration, it is a Type 1 improper integral.

Since the integral has an infinite discontinuity, it is a Type 2 improper integral.

The integral is a proper integral.

(c)

-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.

-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.

-The integral is a proper integral.

d)

-Since the integral has an infinite interval of integration, it is a Type 1 improper integral.

-Since the integral has an infinite discontinuity, it is a Type 2 improper integral.

-The integral is a proper integral.

asked 2021-11-20

Evaluate the definite integral.

${\int}_{2}^{9}9vdv$

asked 2021-05-16

Factor each expression completely.

a.y+2xy

b.$\mathrm{tan}u+2\mathrm{cos}u\mathrm{tan}u$

a.y+2xy

b.

asked 2021-08-14

Evaluate the integral.

${\int}_{0}^{x}(2x-y)dy$

asked 2021-11-07

Find the indefinite integral and check the result by differentiation.

$\int (2-\frac{3}{{x}^{10}})dx$

asked 2021-08-07

Evaluate the integral.

$\int x{2}^{{x}^{2}}dx$